Contents of this Precalculus tutorial:

Integer, Rational numbers, Real numbers, Complex numbers, Imaginary axis, Imaginary unit, Addition, Multiplication.

What are complex numbers?

Operations on complex numbers

Absolute value of complex numbers, sets on the complex plane

Let’ see what complex numbers are.

First, let’s talk a bit about numbers.

This is 3, for example.

And this is 4.

And unfortunately, sometimes we need negative numbers, too.

Then we may need numbers that express ratios.

These are called rational numbers.

Like the solution of this equation:

And then there are equations where the solution is not a rational number.

So, we introduce the irrational numbers that fill the gap between the rational numbers on the number line.

And that takes us to real numbers. At every point of the number line there is a real number.

But in certain cases - especially if physicists are lurking around - we need numbers that have some quite unusual properties.

For example, one like this:

Right off the top of our heads, we cannot find many numbers that would fit here, because

These strange numbers were named imaginary numbers.

Since the real numbers already took up all spots on the number line, we place the imaginary numbers on an axis perpendicular to it.

The unit of the imaginary axis is .

Its most important property is .

Numbers that consist of real and imaginary parts are called complex numbers.

So, complex numbers are in the form of , and they are located on the so called complex plane.

Here are two complex numbers:

and let’s see how we add or even multiply them together.

For addition, we simply add the real parts

and the imaginary parts.

Multiplication is more exciting.

But .

The funniest is division.

Stay tuned...

Precalculus tutorial

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